Parameter estimators have been used to generate parameters estimates of parameters estimated from an input. Bias estimators have been used to generate bias esitmates of bias of an input. The parameter estimators and bias estimators are methods having many possible applications. One such application is range determination of the Gobal Positioning System (GPS) generating pseudoranges which are affected by potential bias, ionospheric propagation delays and phase ambiguities producing ranging errors. GPS receivers are used to acquire position on ground, in air or in space based upon the reception of satellite signals transmitted from the GPS satellite constellation. From the GPS satellite transmit antenna, the satellite signals propagate through free space, the ionosphere and the troposphere to the GPS receiver. The GPS constellation consists of 24 satellites operating in twelve-hour orbits at an altitude of 20,183 km and provides visibility of six to eleven satellites at elevations of five degrees or more above the horizon to users located anywhere in the world at any time. The navigation signals from the satellites consist of two rf frequencies, L.sub.1 at 1575.42 MHz and L.sub.2 at 1227.6 MHz. The L.sub.1 and L.sub.2 signals are modulated by pseudo-random noise (PN) or spread-spectrum (SS) codes and are also modulated with the navigation data-stream at 50 bps carrying various navigation messages. The signals transmitted from different satellites occupy the same rf bandwidth on a Code Division Multiple Accessing (CDMA) basis. In the CDMA techniques, different PN codes are assigned to different satellites and the receiver matches these codes with like reference codes generated in the receiver through cross-correlation technique implemented in Delay Lock Loops (DLLs). Individual DLLs are assigned to different satellites being tracked by the receiver in a parallel approach or a single loop is shared in a time division mode among many satellites in a single channel and less expansive receivers. In addition to making it possible to separate signals from different GPS satellites, the PN codes also make it possible to measure the range to the satellite by measuring the signal propagation delay from the satellite to the GPS receiver by measuring the relative phase of the received signal code phase with the local reference code phase. The accuracy with which such a propagation delay can be measured depends upon the PN code chip rate, the latter directly determining the rf bandwidth of the spread spectrum modulated signal. The measurement accuracy of the propagation delay is equal to a fraction of the PN code chip period, that is, the inverse of the code chip rate. The actual fraction depends upon the various details of receiver implementation and various sources of errors. Each GPS satellite has two codes assigned to it, a distinct C/A code with a chip rate of 1.023 MHz and a P-Code with different offsets and with a chip rate of 10.23 MHz. The L.sub.1 signal is modulated with both the P and C/A codes in phase quadrature and the L.sub.2 signal is modulated with the P-Code.
From the knowledge of signal propagation delay, the range to any satellite can be obtained by multiplying the delay by the velocity of light in free space if the signal is assumed to propagate in free space and all sources of measurement errors are ignored. However, as the GPS signal propagates through the ionosphere and troposphere of the earth, the delay and velocity of light product is termed pseudo-range and will be only approximately equal to the true geometric range. For the purpose of signal propagation delay measurement, the receiver clock must be in synchronization with the GPS satellite clocks controlling the timing of the transmitted GPS signals. The satellite clocks are highly stable atomic standards and any deviation of their timing from the GPS time is monitored and broadcast by the GPS Control Center to users through clock correction navigation messages. Any residual deviation will contribute to error in the pseudorange measurement. If the receiver had GPS time available to it, then by range measurement to at least three satellites, the receiver can determine its position relative to the GPS satellite by triangulation. Also, with the knowledge of GPS satellite positions in a common coordinate system such as the ECEF (Earth-Centered Earth-Fixed) system, the receiver can determine its position anywhere in the world. The positions of satellites are broadcast through ephemeris navigation messages. The absolute navigation accuracy provided by GPS is a function of the accuracy of the pseudorange measurements. The pseudorange measurements are affected by GPS system errors, such as ephemeris errors, clock errors, atmospheric effects of the ionosphere and troposphere and receiver noise.
Ionospheric and tropospheric effects may be significant. The troposphere is the lower part of the atmosphere extending up to an altitude of about 40 km. The propagation delay of the troposphere reaches about 1.9 to 2.5 meters in the zenith direction and increases approximately with the cosecant of the elevation angle. The tropospheric propagation delay is a function of barometric pressure, temperature, humidity, and other weather variables. However, it is rot a function of signal frequency. The tropospheric delay is modeled by mathematical models and can be computed from these models to high accuracy and the modeling error may be a few decimeters if the water vapor measurement is made with sufficient accuracy. The ionosphere ranges between apaproximately 100 km to 1000 km in altitude and behaves as a dispersive medium. The refractive index is a function of frequency, mainly due to the presence of free electrons. The ionospheric propagation delay is proportional to the total electron count which is equal to the integral of the electron density along the propagation path of the signal. Due to the dispersive nature of the medium, the ionosphere introduces a phase delay in the carrier phase and a group delay in the code phase which are equal and opposite in sign, i.e., the carrier phase is advanced while the code phase is delayed. The code delay is also inversely proportional to the square of the carrier frequency of the rf signal to a high approximation and is also a function of the elevation angle via the obliquity factor. In mid-latitude regions, the ionospheric delay can vary between 8 to 30 meters, depending upon the time of day, with possibly much higher values in the equatorial and polar regions. During the periods of high sunspot activity, the delays will be much higher and also exhibit faster variations with time. For single frequency GPS users, the ionospheric corrections are broadcast by GPS messages derived on the basis of mathematical models of the ionosphere. Such models compensate for about half of the delay. Thus, the ionospheric error after corrections is of the order of eight meters, one standard deviation, during a normal solar period. For many precision GPS applications, the ionospheric error is the dominant source of error. This error level is high and dual frequency measurements are used to more accurately estimate the ionospheric propagation delay.
The Standard Dual Frequency (SDF) method makes the ionospheric delay estimate on the basis of code phase measurements alone instead of both the code and carrier phase measurements which have relatively much higher noise variances compared to the carrier phase measurements. As such, the ionospheric delay estimation error using the standard dual frequency method is relatively high especially in the presence of moderate to high ionospheric activity.
Using a standard dual frequency estimation method, range is computed using the dual frequency code measurements providing two equations and two unknown so that and estimates for the ionosphere delays can be computed. Ionospheric delay estimation is conveniently described by a mathematical development. Estimating ionospheric propagation delay from noisy C/A and P/Y code signals, and the L1 and L2 carrier phase measurements may be formulated in terms of the following range equations. ##EQU1##
Range .rho. is the geometric distance between the GPS satellite transmitter and the GPS receiver, plus non-dispersive contributions such as tropospheric refraction and clock drift. The I/f.sup.2 terms represent ionospheric propagation delays where I is proportional to the total electron count of the ionosphere and where f.sub.1 and f.sub.2 are the frequencies having respective wavelengths of .lambda..sub.1 and .lambda..sub.2 for the L.sub.1 and L.sub.2 signals. The range .rho. comprises the delay of free space and the delay of troposphere. From the R.sub.1 and R.sub.2 range equations, range .rho. can be determined. However, such determination does not include the reduction of bias nor the resolution of ambiguities. The terms v.sub.R.sbsb.i and v.sub..phi..sbsb.i for i=1,2 represent receiver noise as manifested in code and cumulative carrier phase measurements respectively, and N.sub.1 and N.sub.2 are the integer ambiguities in the carrier phase measurements.
The range equations may be rewritten as observation equations using an ionospheric delay equation and a coefficient equation.